Calculating The Area Of A Rectangular Sign A Math Problem

Hey guys! Let's dive into a fun math problem today that involves calculating the area of a rectangular sign. This is a practical skill that comes in handy in many real-life situations, whether you're planning a garden, designing a room, or, in this case, figuring out the size of a sign. So, let's break it down step by step and make sure we understand exactly how to tackle this kind of problem.

Understanding the Problem

The problem states that we have a sign that is rectangular in shape. The sign has a width of $2 \frac{1}{3}$ yards and a length of $3 \frac{1}{2}$ yards. Our mission is to find the area of this sign. Remember, the area of a rectangle is found by multiplying its length by its width. So, we need to multiply these two mixed numbers together. But before we jump into the multiplication, let’s make sure we're all on the same page about mixed numbers and how to work with them.

First off, what exactly are mixed numbers? Mixed numbers are numbers that combine a whole number and a fraction, like our $2 \frac{1}{3}$ and $3 \frac{1}{2}$. Dealing with these directly in multiplication can be a bit tricky, so the first thing we're going to do is convert them into improper fractions. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion makes the multiplication process much smoother. Think of it like this: we're changing the way the number looks without actually changing its value. It's like putting on a different outfit – you're still the same person underneath!

To convert a mixed number to an improper fraction, we follow a simple process. We multiply the whole number by the denominator of the fraction, and then we add the numerator. This gives us the new numerator for our improper fraction. The denominator stays the same. Let's walk through this with our numbers. For $2 \frac{1}{3}$, we multiply 2 (the whole number) by 3 (the denominator), which gives us 6. Then, we add 1 (the numerator), giving us 7. So, the improper fraction is $\frac{7}{3}$. We do the same for $3 \frac{1}{2}$. We multiply 3 by 2, which gives us 6, and then add 1, giving us 7. So, the improper fraction is $\frac{7}{2}$. Now that we have our two mixed numbers converted into improper fractions, we're ready to multiply!

Converting Mixed Numbers to Improper Fractions

Alright, let’s get into the nitty-gritty of converting mixed numbers into improper fractions. This is a crucial step because multiplying fractions is much easier when they're in improper form. As we touched on earlier, a mixed number combines a whole number and a fraction, like our example of $2 \frac{1}{3}$ yards for the width of the sign and $3 \frac{1}{2}$ yards for the length. Trying to multiply these directly can lead to a bit of a headache, so we transform them into improper fractions first.

So, how do we do this magical transformation? The process is actually quite straightforward. Let's break it down into clear steps, so it's super easy to follow. The key is to remember that we're essentially figuring out how many equal parts make up the whole number part of our mixed number, and then adding that to the fractional part.

First, we focus on the whole number part of the mixed number. We multiply this whole number by the denominator of the fractional part. This tells us how many fractional parts are contained within the whole number. For example, with $2 \frac{1}{3}$, we multiply 2 (the whole number) by 3 (the denominator). This gives us 6. What this means is that the whole number 2 is equivalent to 6 thirds (since 3 thirds make a whole). Think of it like having two pizzas, each cut into three slices. You would have a total of six slices.

Next, we take this result and add it to the numerator of the original fraction. This is where we're adding the fractional part that was already there. In our example, we add the 6 (from 2 times 3) to the 1 (the numerator of the $\frac{1}{3}$), which gives us 7. This 7 becomes the new numerator of our improper fraction. We're essentially counting up all the fractional parts we have – the ones from the whole number and the one that was already a fraction.

Finally, we keep the original denominator. The denominator tells us the size of the parts we're counting, and that doesn't change when we convert to an improper fraction. So, in our example, the denominator stays as 3. This means our improper fraction is $\frac{7}{3}$. We've successfully converted $2 \frac{1}{3}$ into an improper fraction!

Let's do the same for the length, $3 \frac{1}{2}$. We multiply the whole number 3 by the denominator 2, which gives us 6. Then, we add the numerator 1, which gives us 7. We keep the original denominator of 2, so our improper fraction is $\frac{7}{2}$.

Now, with both numbers converted to improper fractions, we're set to tackle the multiplication. Remember, the goal here is to make the calculation easier and more straightforward. By converting to improper fractions, we've eliminated the complexity of dealing with mixed numbers directly. It's like preparing your ingredients before you start cooking – it makes the whole process smoother and more enjoyable. So, let's move on to the next step and multiply these fractions together!

Multiplying Improper Fractions

Okay, we've successfully converted our mixed numbers into improper fractions. Now comes the fun part: multiplying them! Multiplying fractions is actually quite straightforward once you get the hang of it. The rule is simple: you multiply the numerators (the top numbers) together to get the new numerator, and you multiply the denominators (the bottom numbers) together to get the new denominator. No need to find common denominators or anything like that – it's just a straight multiplication across.

In our case, we have the fractions $\frac{7}{3}$ (which is the improper fraction equivalent of $2 \frac{1}{3}$) and $\frac{7}{2}$ (which is the improper fraction equivalent of $3 \frac{1}{2}$). To find the area of the sign, we need to multiply these two fractions together.

So, let's start with the numerators. We multiply 7 (the numerator of the first fraction) by 7 (the numerator of the second fraction). This gives us 49. So, the new numerator for our answer will be 49.

Next, we move on to the denominators. We multiply 3 (the denominator of the first fraction) by 2 (the denominator of the second fraction). This gives us 6. So, the new denominator for our answer will be 6.

Putting it all together, we get the fraction $\frac{49}{6}$. This is the area of the sign in square yards, expressed as an improper fraction. But wait, we're not quite done yet! While $\frac{49}{6}$ is a perfectly valid answer, it's usually more helpful to express it as a mixed number. This makes it easier to visualize the size of the area. Think about it – it's easier to imagine 8 and a bit square yards than 49 sixths of a square yard.

So, the next step is to convert this improper fraction back into a mixed number. This involves dividing the numerator by the denominator and seeing what we get. Let's dive into that conversion process next.

Converting Back to a Mixed Number

Alright, we've arrived at an improper fraction, $\frac{49}{6}$, which represents the area of our sign in square yards. But, as we discussed, it’s often more practical and easier to understand this area if we convert it back into a mixed number. This involves figuring out how many whole numbers and how much of a fraction we have. It’s like taking a pile of building blocks and seeing how many complete towers you can build, and how many blocks are left over.

The process is pretty straightforward: we perform division. Specifically, we divide the numerator (49) by the denominator (6). This division will tell us how many whole times 6 fits into 49, and what the remainder is. The whole number part of the answer will be the whole number in our mixed number, and the remainder will help us determine the fractional part.

So, let’s do the division: 49 divided by 6. How many times does 6 go into 49? Well, 6 times 8 is 48, which is the closest we can get to 49 without going over. So, 6 goes into 49 eight times. This means the whole number part of our mixed number is 8. We have 8 whole square yards in the area of our sign.

But we’re not done yet! We have a remainder to deal with. The remainder is the amount left over after we’ve divided as many whole times as we can. In this case, we subtracted 48 (6 times 8) from 49, which leaves us with a remainder of 1. This remainder becomes the numerator of our fractional part.

The denominator of our fractional part stays the same as the denominator of our original improper fraction, which is 6. So, our fractional part is $\frac{1}{6}$. This means we have an additional one-sixth of a square yard in the area of our sign.

Putting it all together, our mixed number is $8 \frac{1}{6}$. This tells us that the area of the rectangular sign is 8 and one-sixth square yards. This is much easier to visualize and understand than the improper fraction $\frac{49}{6}$. It’s like saying you have 8 whole pizzas and one-sixth of another pizza, rather than saying you have 49 slices when each pizza is cut into 6 slices.

Now that we have our final answer in a clear and understandable form, let’s circle back to the original question and see how our answer aligns with the options provided.

Matching the Answer to the Options

Okay, we've done the math, we've converted fractions, and we've arrived at our answer: the area of the sign is $8 \frac{1}{6}$ square yards. Now, the final step is to match our answer to the options given in the problem. This is a crucial step because it ensures we haven't made any silly mistakes along the way and that we're selecting the correct choice.

The original problem provided us with four options:

• A. $8 \frac{1}{6}$ square yards
• B. $6 \frac{2}{6}$ square yards
• C. $6 \frac{1}{6}$ square yards
• D. $8 \frac{2}{6}$ square yards

Looking at these options, it's clear that our calculated answer, $8 \frac{1}{6}$ square yards, matches perfectly with option A. This is great news! It confirms that we've followed the correct steps and haven't made any calculation errors. It's always satisfying when your answer lines up neatly with one of the choices provided.

But, just for the sake of being thorough, let's quickly glance at the other options and see why they're not the correct answer. Option B, $6 \frac{2}{6}$ square yards, and option C, $6 \frac{1}{6}$ square yards, are both significantly smaller than our calculated area. This suggests there might have been an error in multiplying the fractions or in converting the mixed numbers if someone had arrived at these answers. Option D, $8 \frac{2}{6}$ square yards, is close to our answer, but the fractional part is different. This could indicate a mistake in the final conversion back to a mixed number, perhaps miscalculating the remainder.

However, since our answer matches option A perfectly, we can confidently select this as the correct answer. We've successfully navigated the problem, converted mixed numbers to improper fractions, multiplied those fractions, converted back to a mixed number, and matched our answer to the given options. That’s a lot of math in one go!

So, to wrap it all up, the area of the rectangular sign is indeed $8 \frac{1}{6}$ square yards. Great job, guys! You've conquered this math problem, and you're one step closer to mastering the art of fraction calculations. Remember, practice makes perfect, so keep tackling these kinds of problems, and you'll become a fraction-multiplying pro in no time!

Final Answer

Therefore, the final answer is A. $8 \frac{1}{6}$ square yards